I made this point in a comment at Not Even Wrong, but I think it worth amplifying. It is regarding section 2.3 of a recent document released by Mochizuki, which I reproduce below the fold, along with my discussion. I have taken a screenshot as I don’t know if the document will be updated or not. I’m trying to figure this out as I go, it’s just so bizarre [Edit as in, I was literally thinking it through as I wrote this, and I didn’t edit it, on purpose. It’s what David Butler tags as #trymathslive on Twitter].
Here is the relevant section. My discussion below will be performed using standard notation and terminology, which clearly doesn’t affect my conclusions, and makes things easier to read.
The argument in (ii) seems to me to be the following: if we work in a skeleton of the category of topological spaces, where all one-element topological spaces are actually equal (because there is only one) then the interval is collapsed to the circle . But doesn’t this argument also imply that every topological space is collapsed to a single point? Because, given a topological space with at least two points , we just run the argument in (ii) with the obvious substitutions, so that and are “identified” (I will explain the quotation marks below). But and are arbitrary, so that all pairs of points are “identified”, and so is replaced by the (unique) single element space, the quotient by the resulting equivalence relation. As a result, every non-empty topological space is isomorphic to every other one, in particular to the one-element space. This would seem to imply that the skeleton of the category of topological spaces collapses to the interval category with two objects, and one non-identity arrow.
The only way I can try to resolve this is that there is a purely linguistic sleight of hand here. Clearly working in a skeleton of the category of topological spaces in no way forces all non-empty topological spaces to collapse. But the use of the word “identified” here is ambiguous, and has two meanings in the current discussion:
- We can say in linear algebra that we identify any real -dimensional vector space with , meaning that up to isomorphism, we can assume we are working with the latter. There may be some minor details that need care, like whether the canonical basis of is being implicitly used or not, compared to a generic vector space without a chosen basis. This is mathematical, and well-known.
- But we can also say that given a topological space, we identify a pair of points in that space, meaning that we perform a quotient construction. This changes the space we are working with completely, resulting with something not isomorphic to the original (this is the point of 2.3.ii), ignoring the issue of skeleta).
But the verb “identify” here is overloaded, and doing different things in each situation. This is obvious to any mathematician. And the complaint of Mochizuki is that his critics, what he terms “the RCS” (namely, Peter Scholze and Jakob Stix, I don’t know why he doesn’t do them the courtesy of naming them), are doing the former, but he is pointing to the latter as giving rise to some kind of problem. But this metaphor is at best a metaphor: it is not “closely technically related” to IUT, since i) and ii) are dealing with different meanings of the word, applying in different situations. The indented sentence starting “it is by no means…” is completely irrelevant, and I feel confused, since the sense in which and are identified as being isomorphic in the first half of the sentence is completely different from the sense in which they are identified as being equal points in a quotient space in the second sentence.
If Mochizuki’s writing was less circumlocutory (why do we need fancy notation for basic objects? and reminders of how they are built?), then it would be easier to grasp these apparently illustrative and enlightening examples. But it’s taken me several readings to understand that this seems to be a metaphor, at best, rather than a blatantly false statement. But the metaphor is being conflated with the situation it is meant to be explaining, due to a quirk of terminology. To quote the man himself from section 2.2 of the document “In fact, of course, such “pseudo-mathematical reasoning” is itself fundamentally flawed. [sic]”. I don’t find the pseudo-explanations involving line segments as metaphors for morphisms in a category at all useful, and indeed I find them actively misleading for people who aren’t familiar enough with category theory to take the metaphor of 2. above as being a faithful representation of the situation in 1.
Added: There was some discussion on Reddit about this post. I made some responses around points that I realised I failed I made clearly enough, and now I think my issue with the above example is that it is a total straw man. Part i) could have been written without reference to or the specific subspaces and , and it’s really just talking about terminal objects in a category, and the possibility that one can assume there is a unique terminal object, as long as one is willing to replace the category one is working with an equivalent one (you don’t even need to pass to a skeleton). Reusing the notation and terminology from i) in ii) seems chosen to conflate the two in the mind of the casual reader (it did for me, and I know better!). The indented sentence could be phrased more accurately and transparently as
“the fact all terminal objects in a category are uniquely isomorphic to each other doesn’t imply taking a coequaliser won’t give you a new object”
which is true, but tells us nothing about the veracity of Scholze and Stix’s argument.